Optimized quantum sensing with a single electron spin using real-time adaptive measurements C. Bonato1,†, M.S. Blok1, †, H. T. Dinani2,3, D. W. Berry2, M. L. Markham4, D. J. Twitchen4, R. Hanson1,* 1 QuTech and Kavli Institute of Nanoscience, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands 2 Department of Physics and Astronomy, Macquarie University, Sydney, New South Wales 2109, Australia 3 Center for Engineered Quantum Systems, Macquarie University, Sydney, New South Wales 2109, Australia 4 Element Six Ltd, Kings Ride Park, Ascot, Berkshire SL5 8BP, UK, † These authors contributed equally to this work * corresponding author:
[email protected] Quantum sensors based on single solid-state spins promise a unique combination of sensitivity and spatial resolution1-20. The key challenge in sensing is to achieve minimum estimation uncertainty within a given time and with a high dynamic range. Adaptive strategies have been proposed to achieve optimal performance but their implementation in solid-state systems has been hindered by the demanding experimental requirements. Here we realize adaptive d.c. sensing by combining single-shot readout of an electron spin in diamond with fast feedback. By adapting the spin readout basis in real time based on previous outcomes we demonstrate a sensitivity in Ramsey interferometry surpassing the standard measurement limit. Furthermore, we find by simulations and experiments that adaptive protocols offer a distinctive advantage over the best-known non-adaptive protocols when overhead and limited estimation time are taken into account. Using an optimized adaptive protocol we achieve a magnetic field sensitivity of 6.1 ± 1.7 nT Hz-1/2 over a wide range of 1.78 mT. These results open up a new class of experiments for solidstate sensors in which real-time knowledge of the measurement history is exploited to obtain optimal performance. Quantum sensors have the potential to achieve unprecedented sensitivity by exploiting control over individual quantum systems1,2. As a prominent example, sensors based on single electron spins associated with Nitrogen-Vacancy (NV) centers in diamond capitalize on the spin’s quantum coherence and the high spatial resolution resulting from the atomic-like electronic wave function3,4. Pioneering experiments have already demonstrated single-spin sensing of magnetic fields5-7, electric fields8, temperature9,10 and strain11. NV sensors may therefore have a revolutionary impact on biology1215 , nanotechnology16-18 and material science19,20. A spin-based magnetometer can sense a d.c. magnetic field through the Zeeman shift = ℏ = ℏ2 ( is the gyromagnetic ratio and the Larmor frequency) between two spin levels |0 and |1 . In a Ramsey interferometry
experiment, a superposition state 1/√2 |0 + |1 , prepared by a π/2 pulse, will evolve to 1/√2 |0 + |1 over a sensing time t. The phase = 2 can be measured by reading out the spin in a suitable basis, by adjusting the phase ϑ of a second π/2 pulse. For a Ramsey experiment that is repeated with constant sensing time the uncertainty decreases with the total sensing time T as (standard measurement sensitivity, 1/ 2 √ SMS). However, the field range also decreases with because the signal is periodic, creating ambiguity whenever |2 | > . This results in a dynamic range bounded as ≤ % / . ,!"# / Recently, it was discovered that the use of multiple sensing times within an estimation sequence can yield a scaling of proportional to 1/ , resulting in a vastly improved dynamic range: ,!"# / ≤
/'! ( , where '! ( is the shortest sensing time used. A major open question is whether adaptive protocols, in which the readout basis is optimized in real time based on previous outcomes, can outperform non-adaptive protocols. While scaling beating the standard measurement limit has been reported with non-adaptive protocols22,23, feedback techniques have only recently been demonstrated for solid-state quantum systems24-26 and adaptive sensing protocols have so far remained out of reach. Here we implement adaptive d.c. sensing with a single-electron spin magnetometer in diamond by exploiting high-fidelity single-shot readout and fast feedback electronics (Fig. 1a). We demonstrate a sensitivity beyond the standard measurement limit over a large field range. Furthermore, we investigate through experiments and simulations the performance of different adaptive protocols and compare these to the best known non-adaptive protocol. Although the non-adaptive protocol improves on the standard measurement limit for sequences with many detections we find that the adaptive protocols perform better when overhead time for initialization and readout is taken into account. In particular, the adaptive protocols require shorter sequences to reach the same sensitivity, thus allowing for sensing of signals that fluctuate on shorter timescales. Our magnetometer employs two spin levels of a single NV center electron in isotopically purified diamond (0.01% 13C). We exploit resonant spinselective optical excitation, at a temperature of 8 K, for high-fidelity initialization and single-shot readout27 (Fig. 1b). Microwave pulses, applied via an on-chip stripline, coherently control the electron spin state. From Ramsey experiments, we measure a spin dephasing time of )∗ = 96 ± 2 μs (Fig. 1c). In order to characterize the performance of different sensing protocols in a controlled setting, the effect of the external field is implemented as an artificial frequency detuning, by adding = 2 to the phase . of the final π/2 pulse. To achieve high sensitivity in a wide field range we use an estimation sequence consisting of N different sensing times21-23,28, varying as '( = 2/0( '! ( 1 = 1. . 2 . The value of '! ( sets the range; we take τmin= 20 ns, corresponding to a range | | < 25 MHz, equivalent to |B|0. For each Ramsey run, in the case (9=5, :=2), the time taken by the microprocessor to perform the Bayesian update ranges between 80μs and 190μs. This time is comparable to the spin initialization duration, so both operations can be performed simultaneously, with no additional overhead (Supplementary Table 4).
Acknowledgements We thank Marijn Tiggelman and Raymond Schouten for the development of the FPGA. We acknowledge support from the Dutch Organization for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO), the DARPA QuASAR programme, the EU SOLID, and DIAMANT programmes and the European Research Council through a Starting Grant. D.W.B. is funded by an Australian Research Council Future Fellowship (FT100100761).
References [1] Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nature Photonics 5, 222229 (2011). [2] Higgins, B. L., et al. Entanglement-free Heisenberg-limited phase estimation. Nature 450, 393 (2007). [3] Degen, C. L. Scanning magnetic field microscope with a diamond single-spin sensor. Applied Physics Letters 92, 243111 (2008). [4] Taylor, J. M. et al. High-sensitivity diamond magnetometer with nanoscale resolution. Nature Physics 4, 810-816 (2008). [5] Maze, J. R. et al. Nanoscale magnetic sensing with an individual electronic spin in diamond. Nature 455, 644-647 (2008). [6] Balasubramanian, G., et al. Nanoscale imaging magnetometry with diamond spins under ambient conditions. Nature 455, 648 (2008). [7] Balasubramanian, G., et al. Ultralong spin coherence time in isotopically engineered diamond. Nature Materials 8, 383-387 (2009). [8] Dolde, F. et al. Electric-field sensing using single diamond spins. Nature Physics 7, 459-453 (2011). [9] Acosta, V. M., et al. Temperature dependence of the nitrogen-vacancy magnetic resonance in diamond. Physical Review Letters 104, 070801 (2010). [10] Toyli, D. M. et al. Fluorescence thermometry enhanced by the quantum coherence of single spins in diamond. Proc. Natl. Acad. Sci.110, 8417 (2013). [11] Ovartchaiyapong, P., Lee, K. W., Myers, B. A. & Bleszynski Jayich, A. C. Dynamic strain-mediated coupling of a single diamond spin to a mechanical resonator. Nature Communications 5, 4429 (2014). [12] Le Sage, D. et al. Optical magnetic imaging of living cells. Nature 496, 486-489 (2013). [13] Kaufmann S., et al. Detection of atomic spin labels in a lipid bilayer using a single-spin nanodiamond probe. Proc. Natl. Acad. Sci. 110, 10894 (2013). [14] Kucsko, G. et al. Nanometre-scale thermometry in a living cell. Nature 500, 54-58 (2013). [15] Shi, F. et al. Single-protein spin resonance spectroscopy under ambient conditions. Science 347, 1135 (2015). [16] Maletinsky, P. et al. A robust scanning diamond sensor for nanoscale imaging with single nitrogen-vacancy centres. Nature Nanotechnology 7, 320-324 (2011). [17] Staudacher, T. et al. Nuclear Magnetic Resonance Spectroscopy on a (5-Nanometer)3 Sample Volume. Science 339, 561 (2013). [18] Mamin, H. J. et al. Nanoscale Nuclear Magnetic Resonance with a Nitrogen-Vacancy Spin Sensor. Science 339, 557 (2013). [19] Tetienne, J.-P. et al. Nanoscale imaging and control of domain-wall hopping with a nitrogenvacancy center microscope. Science 344, 1366 (2014). [20] Kolkowitz, S. et al. Probing Johnson noise and ballistic transport in normal metals with a singlespin qubit. Science 347, 1129 (2015). [21] Said R. S., Berry D. W. & Twamley, J. Nanoscale magnetometry using a single-spin system in diamond. Physical Review B 83, 125410 (2011). [22] Waldherr, G. et al. High-dynamic-range magnetometry with a single nuclear spin in diamond. Nature Nanotechnology 7, 105 (2012). [23] Nusran, N. M. et al. High-dynamic-range magnetometry with a single electronic spin in diamond. Nature Nanotech. 7, 109 (2012). [24] Vijay, R., et al. Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback. Nature 490, 77 (2012).
[25] Blok, M. S. et al. Manipulating a qubit through the backaction of sequential partial measurements and real-time feedback. Nature Physics 10, 189 (2014). [26] Shulman, M. D., et al. Suppressing qubit dephasing using real-time Hamiltonian estimation. Nature Communications 5, 5156 (2014) [27] Robledo, L., et al. High-fidelity projective read-out of a solid-state spin quantum register. Nature 477, 574 (2011). [28] Cappellaro, P. Spin-bath narrowing with adaptive parameter estimation. Physical Review A 85, 030301 (2012). [29] Hentschel, A. & Sanders, B. C. Machine Learning for Precise Quantum Measurement, Physical Review Letters 104, 063603 (2010). [30] Hayes, J. F. A. & Berry, D. W. Swarm optimization for adaptive phase measurements with low visibility. Physical Review A 89, 013838 (2014).
Figure 1. Experiment concept and apparatus. (a) The adaptive frequency estimation protocol consists of a sequence of initialization, sensing, measurement operations. After each measurement run, the outcome μ is used to update the estimate of the frequency , which is then used to optimize the sensing parameters for the following run. Experimentally, the frequency estimation and adaptive calculation of the phase are performed in real-time by a microprocessor. (b) The experiment is performed using the states |0〉 = |RO = 0〉, |1〉 = |RO = −1〉 of the electronic spin of a NV centre in diamond. The electronic spin is readout by resonant optical excitation and photon counting27: detection of luminescence photons corresponds to detection of the |0〉 state. We plot the probability of detecting a photon after initializing either in |0〉 or |1〉. The readout fidelities for the states |0〉 (outcome 0) and |1〉 (outcome 1) are :\ = 0.88 ± 0.02, :Z = 0.98 ± 0.02, respectively. (c) Each measurement run consists of a Ramsey experiment, in which the phase accumulated over time by a spin superposition during free evolution is measured. The measurement basis rotation is controlled by the phase ϑ of the final π/2 pulse. From the measured phase, we can extract the frequency , corresponding to an energy shift between the levels |0〉and |1〉 given by an external field (magnetic field, temperature, strain…). Here, to test the performance of different protocols, we set as an artificial detuning, set by the microprocessor by adding = 2 to the phase ϑ (Supplementary Figure S7).
Figure 2. High dynamic-range adaptive magnetometry Limited-adaptive protocol, in the case of one Ramsey experiment per sensing time (G=1, F=0). In each step, the current frequency probability distribution 7 is plotted (solid black line), together with conditional probabilities 7 V| for the measurement outcomes V = 0 (red shaded area) and V = 1 (blue shaded area). After each measurement, 7 is updated according to Bayes’ rule. The detection phase ϑ of the Ramsey experiment is set to the angle which attains the best distinguishability between peaks in the current frequency probability distribution 7 . Ultimately, the protocol converges to a single peak in the probability distribution, which delivers the frequency estimate.
Figure 3. Frequency dependence of uncertainty. (a)-(b) Frequency estimate example, for (G=5, F=7). We set a fixed artificial detuning = 2 MHz and run different instances of the limited-adaptive is averaged frequency estimation protocol, with increasing N. The resulting probability density 7 over 101 repetitions. (c) Holevo variance as a function of the frequency for N=2, 4 (limitedadaptive protocol, G=5, F=7). We vary by adjusting the phase of the final π/2 pulse. Solid lines correspond to numerical simulations, performed with 101 repetitions per frequency point and experimental parameters for fidelity and dephasing. Experimental points (triangular shape), were acquired with 101 repetitions each. Error bars (one standard deviation) are calculated by bootstrap analysis.
Figure 4. Scaling of sensitivity as a function of total time. (a) The three protocols are compared by plotting Q ) = BC as a function of the total sensing time T (not including spin initialization and readout). For (G=5, F=2) the non-adaptive protocol (green triangles) is bound to the SMS limit, while for both the limited-adaptive (orange circles) and the optimized adaptive (red triangles) protocols Q ) scales close to 1/ . The sensitivity of the limited-adaptive protocol is, however, worse than the optimized-adaptive one. When increasing the number of Ramsey experiments per sensing time to (G=5, F=7), the non-adaptive protocol (blue triangles) reaches Heisenberg-like scaling, with a sensitivity comparable to the optimized adaptive protocol for (G=5, F=2). (b) By including spin initialization and readout durations, the superiority of the optimized adaptive protocol (red triangles), which requires less Ramsey runs per sensing time (smaller F, G) to reach 1/ scaling, is evidenced. The optimized adaptive protocol can estimate magnetic fields with a repetition rate of 20Hz, with a sensitivity more than one order of magnitude better than the non-adaptive protocol. All data are taken with 700 repetitions per data-point. In both plots, error bars corresponding to one standard deviation of the results are obtained using the bootstrap method.
Supplementary Information Optimized quantum sensing with a single electron spin using real-time adaptive measurements C. Bonato, M.S. Blok, H. T. Dinani, D. W. Berry, M. L. Markham, D. J. Twitchen, R. Hanson
I. Comparison of sensing protocols: numerical simulations In the following, the performances of different single-qubit frequency estimation protocols will be compared through numerical simulations. We will describe and analyse three main sensing algorithms, defined using a pseudo-code in Supplementary Tables 1, 2, 3: the limited-adaptive, nonadaptive and optimized-adaptive protocols. In order to achieve high dynamic range, each estimation sequence consists of N different sensing times, multiples of the shortest sensing time = 20 ns: = 2 ( = 1. . ). After each Ramsey, the electron spin is measured: the result of each detection is indicated in the pseudo-code by the Ramsey (θ, τ) function. The input parameters of this function are the sensing time τ and the phase ϑ of the second π/2 pulse. In the simulation code, this function generates a random value ( = 0, 1), using the python library numpy.random, chosen according to the probability distribution [p0, p1=1-p0], where: =
0|
! # "∗#
+
=
cos(2)
+ *+
(Eq. S-E1)
, , , are, respectively the readout fidelities for ms=0 and ms=1. In the following simulations we use the values: , = 0, 0.75, 0.88 or 1.00 (-. = 0), , = 0.993 (-. = 1), / ∗ = 5μs or 96 μs.
For each Ramsey experiment (indexed here by the label ℓ), the detection result ℓ is used to update | … ℓ ~ | … ℓ . the estimation of the magnetic field using Bayes rule: ℓ| This is indicated in the pseudo-code by the function Bayesian_update (res, θ, τ). Due to its periodicity it is convenient to express P( ) in a Fourier series, resulting in the following update rule: ℓ
3
=
4ℓ
ℓ
3
+
!
5 ∗6 " #
#
7
8
9 ℓ : ;ℓ
ℓ
3
